The A-theory characteristic of a fibration is a
map to Waldhausen's algebraic K-theory of spaces which
can be regarded as a parametrized Euler characteristic of
the fibers. Regarding the classifying space of the cobordism
category as a moduli space of smooth manifolds, stable under
extensions by cobordisms, it is natural to ask whether the
A-theory characteristic can be extended to the cobordism
category. A candidate such extension was proposed by Bökstedt
and Madsen who defined an infinite loop map from the d-dimensional
cobordism category to the algebraic K-theory of BO(d). I will
discuss the connections between this map, the A-theory
characteristic and the smooth Riemann-Roch theorem of Dwyer,
Weiss and Williams.